我們研究一些微分算子的週期譜問題，特別是在無窮多邊形圖上的 Schr"{o}dinger 算子。利用 Floquet-Bloch 理論，我們可以推導及分析由二維長方形構成的週期量子圖之色散關係並推廣到 $n$ 維立方體週期量子圖上。微分算子的譜是由解析簇，或稱 Bloch 簇，所得到。此外，在平面上廣為人知的 11 種阿基米德鑲嵌中，我們選擇其中兩種：截角正方形鑲嵌 (頂點組態為(4,$8^2$)) 及小斜方截半六邊形鑲嵌 (頂點組態為(3,4,6,4)) 來研究，利用系統化的特徵函數方法，一樣可以推導出分別由這兩種鑲嵌組成的量子圖之色散關係。這些鑲嵌的色散關係出乎意料地簡單，讓後續的分析變得可行。

實數線上 Schr"{o}dinger 算子的週期譜研究傳統上是利用 Hill 判別式。Binding 和 Volkmer 近期利用 Pr"{u}fer 幅角巧妙地給出另一種證明。我們將該結果推廣應用於 Atkinson 半正則 $p$-拉普拉斯算子問題。

We study the periodic spectrum of some differential operators, in particular the Schr"{o}dinger operator acting on infinite polygonal graphs. Using Floquet-Bloch theory, we derive and analyze on the dispersion relations of the periodic quantum graph generated by 2-dimensional rectangles, and also $n$-cubes. The analytic variety, also called Bloch variety, gives the spectrum of the differential operators. Furthermore, it is well known that there are 11 types of Archimedean tilings in the plane. We take two of them, the truncated square tiling ((4,$8^2$) in vertex configuration) and the rhombi-trihexagonal tiling ((3,4,6,4) in vertex configuration). Through a systematic characteristic function method, we are able to derive the dispersion relations for the graphs formed by these tilings. We note that these dispersion relations are surprisingly simple, making it possible for further analysis.

Traditionally the periodic spectrum of the Schr"{o}dinger operator on $mathbb{R}$ is studied using Hill's discriminant. Recently, Binging and Volkmer gave an alternative proof with a clever argument using the Pr"{u}fer angle. We generalize their result to the Atkinson's semidefinite $p$-Laplacian operator.

Contents
1 Introduction 1
1.1 Periodic quantum graphs . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Quantum graph of Archimedean tilings . . . . . . . . . . . . . . . . . 8
1.3 Periodic and antiperiodic eigenvalues . . . . . . . . . . . . . . . . . . 11
1.4 Chapter summaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Dispersion relation for periodic n-cubic quantum graphs 14
2.1 Dispersion relation for periodic rectangular graphs . . . . . . . . . . . 14
2.2 Dispersion relation for periodic cubic graphs . . . . . . . . . . . . . . 19
2.3 Dispersion relation for periodic n-cubic graphs . . . . . . . . . . . . . 24
3 Periodic quantum graphs from Archimedean tilings 34
3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2 Dispersion relation for truncated square tiling . . . . . . . . . . . . . 35
3.3 Dispersion relation for rhombi-trihexagonal tiling . . . . . . . . . . . 42
4 Eigenvalues on periodic and antiperiodic p-Laplacian eigenvalue
problems 48
Appendix A Archimedean tiling 55
A.1 Proof of Fyodorov Theorem . . . . . . . . . . . . . . . . . . . . . . . 55
A.2 Proof on Archimedean tilings . . . . . . . . . . . . . . . . . . . . . . 57
Appendix B Computation of some characteristic functions 60
B.1 Characteristic functions for truncated square tiling . . . . . . . . . . 60
B.2 Characteristic functions for rhombi-trihexagonal tiling . . . . . . . . . 62
Appendix C Floquet-Bloch theory 63
List of Figures
1.1 Fundamental domain of general periodic hexagonal graphs . . . . . . 4
2.1 Fundamental domain of periodic rectangular graphs . . . . . . . . . . 14
2.2 Floquet-Bloch conditions on periodic rectangular graphs . . . . . . . 15
2.3 Fundamental domain of periodic cubic graphs . . . . . . . . . . . . . 20
2.4 Floquet-Bloch conditions on periodic cubic graphs . . . . . . . . . . . 21
2.5 Characteristic function for S . . . . . . . . . . . . . . . . . . . . . . . 32
3.1 Fundamental domain of truncated square tiling . . . . . . . . . . . . 35
3.2 Floquet-Bloch conditions on truncated square tiling . . . . . . . . . . 36
3.3 Fundamental domain of rhombi-trihexagonal tiling . . . . . . . . . . . 42
3.4 Floquet-Bloch conditions on rhombi-trihexagonal tiling . . . . . . . . 43
A.1 17 choices for regular polygons to tting around a vertex on a plane . 57
List of Tables
A.1 Table of 11 Archimedean tilings . . . . . . . . . . . . . . . . . . . . . 59

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